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| Mirrors > Home > MPE Home > Th. List > latnlej1l | Structured version Visualization version GIF version | ||
| Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.) |
| Ref | Expression |
|---|---|
| latlej.b | β’ π΅ = (BaseβπΎ) |
| latlej.l | β’ β€ = (leβπΎ) |
| latlej.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latnlej1l | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latlej.b | . . 3 β’ π΅ = (BaseβπΎ) | |
| 2 | latlej.l | . . 3 β’ β€ = (leβπΎ) | |
| 3 | latlej.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 4 | 1, 2, 3 | latnlej 18447 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β (π β π β§ π β π)) |
| 5 | 4 | simpld 493 | 1 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 class class class wbr 5143 βcfv 6543 (class class class)co 7416 Basecbs 17179 lecple 17239 joincjn 18302 Latclat 18422 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-lub 18337 df-join 18339 df-lat 18423 |
| This theorem is referenced by: atnlej1 38908 3atlem4 39015 3atlem6 39017 dalemcnes 39179 lhpexle3lem 39540 cdlemd4 39730 cdlemd7 39733 cdleme0e 39746 cdleme3e 39761 cdleme9 39782 cdleme17c 39817 |
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